Change event type after meeting is scheduled?

We have 5 event types that our primary customer can schedule. It is not unusual for a customer to accidentally book the wrong type of event. We do not know this happens until we are in the scheduled call with them. It seems as though we cannot change the event type for a meeting that has been scheduled. The only work around for this we can find is to create a new meeting under the new event type and copy and past the information from the old meeting and then delete the old meeting.  However, when we’ve tried this people miss the new invite and in turn miss their meeting. We’ve also tried renaming our event types to be more clear, but that hasn’t helped either. We use the even type for analysis of our customer onboarding process, so capturing the correct event type is helpful for us internally.

Is there a way to do this and we’re missing it? 

If not, it would be very useful for us to be able to change a meeting from one event type to another already created event type.